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RowBox[{"k", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", "k"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "k"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 3", LeftTee[ LeftTee[ LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`s], LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k] == LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[ LeftTee[$CellContext`s, $CellContext`s], LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]]], \ $CellContext`k]]], LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]]}], "Tooltip"]& ], TagBox[ TooltipBox[ DiskBox[{-3.4799703967440223`, 6.666666667267577}, 0.19671632147651769`], TagBox[ GridBox[{{"\"Substitution Lemma 4\""}, { RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "s"}], "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "k"}], ")"}], "\[Equal]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"k", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", "k"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "k"}], ")"}]}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"k", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", "k"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "k"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 4", LeftTee[ LeftTee[ LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`s], LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k] == LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]], LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]]}], "Tooltip"]& ], TagBox[ TooltipBox[ DiskBox[{-2.4799703967341316`, 5.83333333385913}, 0.19671632147651769`], TagBox[ GridBox[{{"\"Substitution Lemma 5\""}, { RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{ RowBox[{"s", "\[LeftTee]", "s"}], "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], ")"}]}], "\[LeftTee]", "k"}], ")"}], "\[Equal]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"k", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", "k"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "k"}], ")"}]}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"k", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", "k"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "k"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 5", LeftTee[ LeftTee[ LeftTee[ LeftTee[$CellContext`s, $CellContext`s], LeftTee[$CellContext`s, $CellContext`s]], LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]]], \ $CellContext`k] == LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]], LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]]}], "Tooltip"]& ], TagBox[ TooltipBox[ DiskBox[{-1.4799703967542541`, 5.000000000450683}, 0.19671632147651769`], TagBox[ GridBox[{{"\"Substitution Lemma 6\""}, { RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", "s"}], "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], ")"}]}], ")"}]}], "\[LeftTee]", "k"}], ")"}], "\[Equal]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"k", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", "k"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "k"}], ")"}]}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"k", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", "k"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "k"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 6", LeftTee[ LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]]], LeftTee[ LeftTee[$CellContext`s, $CellContext`s], LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]]]], \ $CellContext`k] == LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]], LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]]}], "Tooltip"]& ], TagBox[ TooltipBox[ DiskBox[{-0.47997039675757946`, 4.166666667042236}, 0.19671632147651769`], TagBox[ GridBox[{{"\"Substitution Lemma 7\""}, { RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "k"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"s", "\[LeftTee]", "s"}], "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], ")"}]}], "\[LeftTee]", "k"}], ")"}]}], ")"}], "\[Equal]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"k", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", "k"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "k"}], ")"}]}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"k", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", "k"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "k"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 7", LeftTee[ LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k], LeftTee[ LeftTee[ LeftTee[$CellContext`s, $CellContext`s], LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]]], \ $CellContext`k]] == LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]], LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]]}], "Tooltip"]& ], TagBox[ TooltipBox[ DiskBox[{0.5200296040812873, 3.3333333336337887`}, 0.19671632147651769`], TagBox[ GridBox[{{"\"Substitution Lemma 8\""}, { RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "k"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", "k"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "k"}], ")"}]}], ")"}]}], ")"}], "\[Equal]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"k", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", "k"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "k"}], ")"}]}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"k", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", "k"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "k"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 8", LeftTee[ LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k], LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]] == LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]], LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]]}], "Tooltip"]& ], TagBox[ TooltipBox[ DiskBox[{1.5200296040308672`, 2.5000000002253415`}, 0.19671632147651769`], TagBox[ GridBox[{{"\"Substitution Lemma 9\""}, { RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"s", "\[LeftTee]", "s"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", "k"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "k"}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"k", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", "k"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "k"}], ")"}]}], ")"}]}], ")"}]}], ")"}], "\[Equal]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"k", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", "k"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "k"}], ")"}]}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"k", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", "k"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "k"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 9", LeftTee[ LeftTee[ LeftTee[$CellContext`s, $CellContext`s], LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]], LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]] == LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]], LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]]}], "Tooltip"]& ], TagBox[ TooltipBox[ DiskBox[{2.5200296041249715`, 1.6666666668168943`}, 0.19671632147651769`], TagBox[ GridBox[{{"\"Substitution Lemma 10\""}, { RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"k", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", "k"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "k"}], ")"}]}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"s", "\[LeftTee]", "k"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "k"}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"k", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", "k"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "k"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}], ")"}], "\[Equal]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"k", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", "k"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "k"}], ")"}]}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"k", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", "k"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "k"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 10", LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]], LeftTee[ LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]], LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]]] == LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]], LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]]}], "Tooltip"]& ], TagBox[ TooltipBox[ DiskBox[{3.520029604096095, 0.8333333334084472}, 0.19671632147651769`], TagBox[ GridBox[{{"\"Substitution Lemma 11\""}, { RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"k", "\[LeftTee]", RowBox[{"(", 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")"}]}], ")"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 3", LeftTee[ LeftTee[ LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`s], LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k] == LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[ LeftTee[$CellContext`s, $CellContext`s], LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]]], \ $CellContext`k]]], LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]]}], "Tooltip"]& ], TagBox[ TooltipBox[ DiskBox[{-3.4799703967440223`, 6.666666667267577}, 0.19671632147651769`], TagBox[ GridBox[{{"\"Substitution Lemma 4\""}, { RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "s"}], "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "k"}], ")"}], "\[Equal]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"k", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", "k"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "k"}], ")"}]}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"k", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", "k"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "k"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 4", LeftTee[ LeftTee[ LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`s], LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k] == LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]], LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]]}], "Tooltip"]& ], TagBox[ TooltipBox[ DiskBox[{-2.4799703967341316`, 5.83333333385913}, 0.19671632147651769`], TagBox[ GridBox[{{"\"Substitution Lemma 5\""}, { RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{ RowBox[{"s", "\[LeftTee]", "s"}], "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], ")"}]}], "\[LeftTee]", "k"}], ")"}], "\[Equal]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"k", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", "k"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "k"}], ")"}]}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"k", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", "k"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "k"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 5", LeftTee[ LeftTee[ LeftTee[ LeftTee[$CellContext`s, $CellContext`s], LeftTee[$CellContext`s, $CellContext`s]], LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]]], \ $CellContext`k] == LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]], LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]]}], "Tooltip"]& ], TagBox[ TooltipBox[ DiskBox[{-1.4799703967542541`, 5.000000000450683}, 0.19671632147651769`], TagBox[ GridBox[{{"\"Substitution Lemma 6\""}, { RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", "s"}], "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], ")"}]}], ")"}]}], "\[LeftTee]", "k"}], ")"}], "\[Equal]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"k", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", "k"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "k"}], ")"}]}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"k", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", "k"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "k"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 6", LeftTee[ LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]]], LeftTee[ LeftTee[$CellContext`s, $CellContext`s], LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]]]], \ $CellContext`k] == LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]], LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]]}], "Tooltip"]& ], TagBox[ TooltipBox[ DiskBox[{-0.47997039675757946`, 4.166666667042236}, 0.19671632147651769`], TagBox[ GridBox[{{"\"Substitution Lemma 7\""}, { RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "k"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"s", "\[LeftTee]", "s"}], "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], ")"}]}], "\[LeftTee]", "k"}], ")"}]}], ")"}], "\[Equal]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"k", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", "k"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "k"}], ")"}]}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"k", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", "k"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "k"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 7", LeftTee[ LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k], LeftTee[ LeftTee[ LeftTee[$CellContext`s, $CellContext`s], LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]]], \ $CellContext`k]] == LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]], LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]]}], "Tooltip"]& ], TagBox[ TooltipBox[ DiskBox[{0.5200296040812873, 3.3333333336337887`}, 0.19671632147651769`], TagBox[ GridBox[{{"\"Substitution Lemma 8\""}, { RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "k"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", "k"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "k"}], ")"}]}], ")"}]}], ")"}], "\[Equal]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"k", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", "k"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "k"}], ")"}]}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"k", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", "k"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "k"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 8", LeftTee[ LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k], LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]] == LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]], LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]]}], "Tooltip"]& ], TagBox[ TooltipBox[ DiskBox[{1.5200296040308672`, 2.5000000002253415`}, 0.19671632147651769`], TagBox[ GridBox[{{"\"Substitution Lemma 9\""}, { RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"s", "\[LeftTee]", "s"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", "k"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "k"}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"k", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", "k"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "k"}], ")"}]}], ")"}]}], ")"}]}], ")"}], "\[Equal]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"k", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", "k"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "k"}], ")"}]}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"k", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", "k"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "k"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 9", LeftTee[ LeftTee[ LeftTee[$CellContext`s, $CellContext`s], LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]], LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]] == LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]], LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]]}], "Tooltip"]& ], TagBox[ TooltipBox[ DiskBox[{2.5200296041249715`, 1.6666666668168943`}, 0.19671632147651769`], TagBox[ GridBox[{{"\"Substitution Lemma 10\""}, { RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"k", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", "k"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "k"}], ")"}]}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"s", "\[LeftTee]", "k"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "k"}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"k", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", "k"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "k"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}], ")"}], "\[Equal]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"k", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", "k"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "k"}], ")"}]}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"k", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", "k"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "k"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 10", LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]], LeftTee[ LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]], LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]]] == LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]], LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]]}], "Tooltip"]& ], TagBox[ TooltipBox[ DiskBox[{3.520029604096095, 0.8333333334084472}, 0.19671632147651769`], TagBox[ GridBox[{{"\"Substitution Lemma 11\""}, { RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"k", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", "k"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", 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RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "k"}], ")"}]}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"k", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", "k"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "k"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 11", LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]], LeftTee[ LeftTee[$CellContext`k, LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, 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"\[LeftTee]", "s"}], "\[LeftTee]", "k"}], ")"}], "\[Equal]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"k", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"s", "\[LeftTee]", "s"}], "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], ")"}]}], "\[LeftTee]", "k"}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"k", "\[LeftTee]", RowBox[{"(", RowBox[{"\[LeftSkeleton]", "1", "\[RightSkeleton]"}], ")"}]}], "\[LeftTee]", RowBox[{"\[LeftSkeleton]", "1", "\[RightSkeleton]"}]}], ")"}]}], ")"}]}], Short[#, 1]& ], "SummaryItem"]}], "\[SpanFromLeft]"}, { RowBox[{ TagBox["\"Axioms: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", TagBox[ TagBox[ RowBox[{ RowBox[{ SubscriptBox["\[ForAll]", RowBox[{"{", RowBox[{"x", ",", "y", ",", "z"}], "}"}]], RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"s", "\[LeftTee]", "x"}], "\[LeftTee]", "y"}], "\[LeftTee]", "z"}], ")"}], "\[Equal]", RowBox[{"(", RowBox[{ RowBox[{"x", "\[LeftTee]", "z"}], "\[LeftTee]", RowBox[{"(", RowBox[{"y", "\[LeftTee]", "z"}], ")"}]}], ")"}]}]}], "&&", RowBox[{ SubscriptBox["\[ForAll]", RowBox[{"{", RowBox[{"x", ",", "y"}], "}"}]], RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"k", "\[LeftTee]", "x"}], "\[LeftTee]", "y"}], ")"}], "\[Equal]", "x"}]}]}], Short[#, 1]& ], "SummaryItem"]}], "\[SpanFromLeft]"}}, GridBoxAlignment -> { "Columns" -> {{Left}}, "Rows" -> {{Automatic}}}, AutoDelete -> False, GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, GridBoxSpacings -> {"Columns" -> {{2}}, "Rows" -> {{Automatic}}}, BaseStyle -> { ShowStringCharacters -> False, NumberMarks -> False, PrintPrecision -> 3, ShowSyntaxStyles -> False}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}, "Rows" -> {{Top}}}, AutoDelete -> False, GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, BaselinePosition -> {1, 1}]}, Dynamic[Typeset`open$$], ImageSize -> Automatic]}, "SummaryPanel"], DynamicModuleValues:>{}], "]"}], ProofObject["EquationalLogic", LeftTee[ LeftTee[ LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]]], $CellContext`s], \ $CellContext`s], $CellContext`k] == LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[ LeftTee[$CellContext`s, $CellContext`s], LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]]], $CellContext`k]]], LeftTee[ LeftTee[$CellContext`k, LeftTee[$CellContext`k, LeftTee[ LeftTee[ LeftTee[$CellContext`s, $CellContext`s], LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]]], $CellContext`k]]], LeftTee[ LeftTee[ LeftTee[$CellContext`s, $CellContext`s], LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k]]]], LeftTee[$CellContext`k, LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k]]]]]]], And[ ForAll[{$CellContext`x, $CellContext`y, $CellContext`z}, LeftTee[ LeftTee[ LeftTee[$CellContext`s, $CellContext`x], $CellContext`y], \ $CellContext`z] == LeftTee[ LeftTee[$CellContext`x, $CellContext`z], LeftTee[$CellContext`y, $CellContext`z]]], ForAll[{$CellContext`x, $CellContext`y}, LeftTee[ LeftTee[$CellContext`k, $CellContext`x], $CellContext`y] == \ $CellContext`x]], {{"Axiom", 1} -> Association["Statement" -> $CellContext`x1 == LeftTee[ LeftTee[$CellContext`k, $CellContext`x1], $CellContext`x2], "Proof" -> Association[]], {"Axiom", 2} -> Association["Statement" -> LeftTee[ LeftTee[$CellContext`x1, $CellContext`x2], LeftTee[$CellContext`x3, $CellContext`x2]] == LeftTee[ LeftTee[ LeftTee[$CellContext`s, $CellContext`x1], $CellContext`x3], \ $CellContext`x2], "Proof" -> Association[]], {"Hypothesis", 1} -> Association["Statement" -> LeftTee[ LeftTee[ LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]]], $CellContext`s], \ $CellContext`s], $CellContext`k] == LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[ LeftTee[$CellContext`s, $CellContext`s], LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]]], $CellContext`k]]], LeftTee[ LeftTee[$CellContext`k, LeftTee[$CellContext`k, LeftTee[ LeftTee[ LeftTee[$CellContext`s, $CellContext`s], LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]]], $CellContext`k]]], LeftTee[ LeftTee[ LeftTee[$CellContext`s, $CellContext`s], LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k]]]], LeftTee[$CellContext`k, LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]]]]], "Proof" -> Association[]], {"SubstitutionLemma", 1} -> Association["Statement" -> LeftTee[ LeftTee[ LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]]], $CellContext`s], \ $CellContext`s], $CellContext`k] == LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[ LeftTee[$CellContext`s, $CellContext`s], LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]]], $CellContext`k]]], LeftTee[$CellContext`k, LeftTee[ LeftTee[ LeftTee[$CellContext`s, $CellContext`s], LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]]], $CellContext`k]]], "Proof" -> Association[ "Input" -> {"Hypothesis", 1}, "Position" -> {2, 1, 2}, "Construct" -> {"Axiom", 1}, "Orientation" -> -1, "Rule" -> (LeftTee[ LeftTee[$CellContext`k, Pattern[$CellContext`x1, Blank[]]], Pattern[$CellContext`x2, Blank[]]] -> $CellContext`x1), "OutputExpression" -> LeftTee[ LeftTee[ LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]]], $CellContext`s], \ $CellContext`s], $CellContext`k] == LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[ LeftTee[$CellContext`s, $CellContext`s], LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]]], $CellContext`k]]], LeftTee[$CellContext`k, LeftTee[ LeftTee[ LeftTee[$CellContext`s, $CellContext`s], LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]]], \ $CellContext`k]]]]], {"SubstitutionLemma", 2} -> Association["Statement" -> LeftTee[ LeftTee[ LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`s], LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k] == LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[ LeftTee[$CellContext`s, $CellContext`s], LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]]], $CellContext`k]]], LeftTee[$CellContext`k, LeftTee[ LeftTee[ LeftTee[$CellContext`s, $CellContext`s], LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]]], $CellContext`k]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 1}, "Position" -> {1, 1}, "Construct" -> {"Axiom", 2}, "Orientation" -> -1, "Rule" -> (LeftTee[ LeftTee[ LeftTee[$CellContext`s, Pattern[$CellContext`x1, Blank[]]], Pattern[$CellContext`x2, Blank[]]], Pattern[$CellContext`x3, Blank[]]] -> LeftTee[ LeftTee[$CellContext`x1, $CellContext`x3], LeftTee[$CellContext`x2, $CellContext`x3]]), "OutputExpression" -> LeftTee[ LeftTee[ LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`s], LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k] == LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[ LeftTee[$CellContext`s, $CellContext`s], LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]]], $CellContext`k]]], LeftTee[$CellContext`k, LeftTee[ LeftTee[ LeftTee[$CellContext`s, $CellContext`s], LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]]], \ $CellContext`k]]]]], {"SubstitutionLemma", 3} -> Association["Statement" -> LeftTee[ LeftTee[ LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`s], LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k] == LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[ LeftTee[$CellContext`s, $CellContext`s], LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]]], $CellContext`k]]], LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 2}, "Position" -> {2, 2, 2}, "Construct" -> {"Axiom", 2}, "Orientation" -> -1, "Rule" -> (LeftTee[ LeftTee[ LeftTee[$CellContext`s, Pattern[$CellContext`x1, Blank[]]], Pattern[$CellContext`x2, Blank[]]], Pattern[$CellContext`x3, Blank[]]] -> LeftTee[ LeftTee[$CellContext`x1, $CellContext`x3], LeftTee[$CellContext`x2, $CellContext`x3]]), "OutputExpression" -> LeftTee[ LeftTee[ LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`s], LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k] == LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[ LeftTee[$CellContext`s, $CellContext`s], LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]]], $CellContext`k]]], LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]]]], {"SubstitutionLemma", 4} -> Association["Statement" -> LeftTee[ LeftTee[ LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`s], LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k] == LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k]]]], LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 3}, "Position" -> {2, 1, 2, 2}, "Construct" -> {"Axiom", 2}, "Orientation" -> -1, "Rule" -> (LeftTee[ LeftTee[ LeftTee[$CellContext`s, Pattern[$CellContext`x1, Blank[]]], Pattern[$CellContext`x2, Blank[]]], Pattern[$CellContext`x3, Blank[]]] -> LeftTee[ LeftTee[$CellContext`x1, $CellContext`x3], LeftTee[$CellContext`x2, $CellContext`x3]]), "OutputExpression" -> LeftTee[ LeftTee[ LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`s], LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k] == LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k]]]], LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]]]], {"SubstitutionLemma", 5} -> Association["Statement" -> LeftTee[ LeftTee[ LeftTee[ LeftTee[$CellContext`s, $CellContext`s], LeftTee[$CellContext`s, $CellContext`s]], LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]]], $CellContext`k] == LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k]]]], LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 4}, "Position" -> {1, 1}, "Construct" -> {"Axiom", 2}, "Orientation" -> -1, "Rule" -> (LeftTee[ LeftTee[ LeftTee[$CellContext`s, Pattern[$CellContext`x1, Blank[]]], Pattern[$CellContext`x2, Blank[]]], Pattern[$CellContext`x3, Blank[]]] -> LeftTee[ LeftTee[$CellContext`x1, $CellContext`x3], LeftTee[$CellContext`x2, $CellContext`x3]]), "OutputExpression" -> LeftTee[ LeftTee[ LeftTee[ LeftTee[$CellContext`s, $CellContext`s], LeftTee[$CellContext`s, $CellContext`s]], LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]]], $CellContext`k] == LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k]]]], LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]]]], {"SubstitutionLemma", 6} -> Association["Statement" -> LeftTee[ LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]]], LeftTee[ LeftTee[$CellContext`s, $CellContext`s], LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]]]], $CellContext`k] == LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k]]]], LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 5}, "Position" -> {1, 1}, "Construct" -> {"Axiom", 2}, "Orientation" -> -1, "Rule" -> (LeftTee[ LeftTee[ LeftTee[$CellContext`s, Pattern[$CellContext`x1, Blank[]]], Pattern[$CellContext`x2, Blank[]]], Pattern[$CellContext`x3, Blank[]]] -> LeftTee[ LeftTee[$CellContext`x1, $CellContext`x3], LeftTee[$CellContext`x2, $CellContext`x3]]), "OutputExpression" -> LeftTee[ LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]]], LeftTee[ LeftTee[$CellContext`s, $CellContext`s], LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]]]], $CellContext`k] == LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k]]]], LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]]]], {"SubstitutionLemma", 7} -> Association["Statement" -> LeftTee[ LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k], LeftTee[ LeftTee[ LeftTee[$CellContext`s, $CellContext`s], LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]]], $CellContext`k]] == LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k]]]], LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 6}, "Position" -> 1, "Construct" -> {"Axiom", 2}, "Orientation" -> -1, "Rule" -> (LeftTee[ LeftTee[ LeftTee[$CellContext`s, Pattern[$CellContext`x1, Blank[]]], Pattern[$CellContext`x2, Blank[]]], Pattern[$CellContext`x3, Blank[]]] -> LeftTee[ LeftTee[$CellContext`x1, $CellContext`x3], LeftTee[$CellContext`x2, $CellContext`x3]]), "OutputExpression" -> LeftTee[ LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k], LeftTee[ LeftTee[ LeftTee[$CellContext`s, $CellContext`s], LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]]], $CellContext`k]] == LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k]]]], LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]]]], {"SubstitutionLemma", 8} -> Association["Statement" -> LeftTee[ LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k], LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k]]] == LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k]]]], LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 7}, "Position" -> {1, 2}, "Construct" -> {"Axiom", 2}, "Orientation" -> -1, "Rule" -> (LeftTee[ LeftTee[ LeftTee[$CellContext`s, Pattern[$CellContext`x1, Blank[]]], Pattern[$CellContext`x2, Blank[]]], Pattern[$CellContext`x3, Blank[]]] -> LeftTee[ LeftTee[$CellContext`x1, $CellContext`x3], LeftTee[$CellContext`x2, $CellContext`x3]]), "OutputExpression" -> LeftTee[ LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k], LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k]]] == LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k]]]], LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]]]], {"SubstitutionLemma", 9} -> Association["Statement" -> LeftTee[ LeftTee[ LeftTee[$CellContext`s, $CellContext`s], LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k]]], LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k]]]] == LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k]]]], LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 8}, "Position" -> 1, "Construct" -> {"Axiom", 2}, "Orientation" -> -1, "Rule" -> (LeftTee[ LeftTee[ LeftTee[$CellContext`s, Pattern[$CellContext`x1, Blank[]]], Pattern[$CellContext`x2, Blank[]]], Pattern[$CellContext`x3, Blank[]]] -> LeftTee[ LeftTee[$CellContext`x1, $CellContext`x3], LeftTee[$CellContext`x2, $CellContext`x3]]), "OutputExpression" -> LeftTee[ LeftTee[ LeftTee[$CellContext`s, $CellContext`s], LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k]]], LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k]]]] == LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k]]]], LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]]]], {"SubstitutionLemma", 10} -> Association["Statement" -> LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k]]]], LeftTee[ LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k]], LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k]]]]] == LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k]]]], LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 9}, "Position" -> 1, "Construct" -> {"Axiom", 2}, "Orientation" -> -1, "Rule" -> (LeftTee[ LeftTee[ LeftTee[$CellContext`s, Pattern[$CellContext`x1, Blank[]]], Pattern[$CellContext`x2, Blank[]]], Pattern[$CellContext`x3, Blank[]]] -> LeftTee[ LeftTee[$CellContext`x1, $CellContext`x3], LeftTee[$CellContext`x2, $CellContext`x3]]), "OutputExpression" -> LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k]]]], LeftTee[ LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k]], LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k]]]]] == LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k]]]], LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]]]], {"SubstitutionLemma", 11} -> Association["Statement" -> LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k]]]], LeftTee[ LeftTee[$CellContext`k, LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k]]]], LeftTee[ LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k], LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k]]]]]] == LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k]]]], LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 10}, "Position" -> {1, 2}, "Construct" -> {"Axiom", 2}, "Orientation" -> -1, "Rule" -> (LeftTee[ LeftTee[ LeftTee[$CellContext`s, Pattern[$CellContext`x1, Blank[]]], Pattern[$CellContext`x2, Blank[]]], Pattern[$CellContext`x3, Blank[]]] -> LeftTee[ LeftTee[$CellContext`x1, $CellContext`x3], LeftTee[$CellContext`x2, $CellContext`x3]]), "OutputExpression" -> LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k]]]], LeftTee[ LeftTee[$CellContext`k, LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k]]]], LeftTee[ LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k], LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]]]] == LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k]]]], LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]]]], {"Conclusion", 1} -> Association[ "Statement" -> True, "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 11}, "Position" -> 1, "Construct" -> {"Axiom", 1}, "Orientation" -> -1, "Rule" -> (LeftTee[ LeftTee[$CellContext`k, Pattern[$CellContext`x1, Blank[]]], Pattern[$CellContext`x2, Blank[]]] -> $CellContext`x1), "OutputExpression" -> True]]}], Editable->False, SelectWithContents->True, Selectable->False]], "Output", TaggingRules->{}, CellChangeTimes->{{3.8086759440378523`*^9, 3.808675968903267*^9}, 3.816015195844708*^9, 3.816017117135301*^9, 3.8160253767617188`*^9, { 3.8161027167262383`*^9, 3.816102778838018*^9}}, CellLabel->"Out[1]=", CellID->375814068] }, Open ]], Cell["Show the abstract proof network:", "Text", TaggingRules->{}, CellChangeTimes->{{3.808675983548875*^9, 3.808675987504202*^9}}, CellID->854433272], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"proof", "[", "\"\\"", "]"}]], "Input", TaggingRules->{}, CellChangeTimes->{{3.8086759650605803`*^9, 3.808675972836635*^9}}, CellLabel->"In[2]:=", CellID->863842568], Cell[BoxData[ GraphicsBox[ NamespaceBox["NetworkGraphics", DynamicModuleBox[{Typeset`graph = HoldComplete[ Graph[{ "Axiom 1", "Axiom 2", "Hypothesis 1", "Substitution Lemma 1", "Substitution Lemma 2", "Substitution Lemma 3", "Substitution Lemma 4", "Substitution Lemma 5", "Substitution Lemma 6", "Substitution Lemma 7", "Substitution Lemma 8", "Substitution Lemma 9", "Substitution Lemma 10", "Substitution Lemma 11", "Conclusion 1"}, {{{ 3, 4}, {1, 4}, {4, 5}, {2, 5}, {5, 6}, {2, 6}, {6, 7}, {2, 7}, {7, 8}, {2, 8}, {8, 9}, {2, 9}, {9, 10}, {2, 10}, {10, 11}, {2, 11}, {11, 12}, {2, 12}, {12, 13}, {2, 13}, {13, 14}, {2, 14}, {14, 15}, {1, 15}}, Null}, { AnnotationRules -> { "Substitution Lemma 10" -> { Tooltip -> Column[{"Substitution Lemma 10", LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]], LeftTee[ LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]], LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]]] == LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]], LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]]}]}, "Substitution Lemma 4" -> { Tooltip -> Column[{"Substitution Lemma 4", LeftTee[ LeftTee[ LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`s], LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k] == LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]], LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]]}]}, "Axiom 1" -> { Tooltip -> Column[{"Axiom 1", $CellContext`x1 == LeftTee[ LeftTee[$CellContext`k, $CellContext`x1], $CellContext`x2]}]}, "Substitution Lemma 2" -> { Tooltip -> Column[{"Substitution Lemma 2", LeftTee[ LeftTee[ LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`s], LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k] == LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[ LeftTee[$CellContext`s, $CellContext`s], LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]]], \ $CellContext`k]]], LeftTee[$CellContext`k, LeftTee[ LeftTee[ LeftTee[$CellContext`s, $CellContext`s], LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]]], \ $CellContext`k]]]}]}, "Substitution Lemma 3" -> { Tooltip -> Column[{"Substitution Lemma 3", LeftTee[ LeftTee[ LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`s], LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k] == LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[ LeftTee[$CellContext`s, $CellContext`s], LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]]], \ $CellContext`k]]], LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]]}]}, "Axiom 2" -> {Tooltip -> Column[{"Axiom 2", LeftTee[ LeftTee[$CellContext`x1, $CellContext`x2], LeftTee[$CellContext`x3, $CellContext`x2]] == LeftTee[ LeftTee[ LeftTee[$CellContext`s, $CellContext`x1], $CellContext`x3], \ $CellContext`x2]}]}, "Substitution Lemma 7" -> { Tooltip -> Column[{"Substitution Lemma 7", LeftTee[ LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k], LeftTee[ LeftTee[ LeftTee[$CellContext`s, $CellContext`s], LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]]], \ $CellContext`k]] == LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]], LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]]}]}, "Substitution Lemma 5" -> { Tooltip -> Column[{"Substitution Lemma 5", LeftTee[ LeftTee[ LeftTee[ LeftTee[$CellContext`s, $CellContext`s], LeftTee[$CellContext`s, $CellContext`s]], LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]]], $CellContext`k] == LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]], LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]]}]}, "Substitution Lemma 8" -> { Tooltip -> Column[{"Substitution Lemma 8", LeftTee[ LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k], LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]] == LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]], LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]]}]}, "Conclusion 1" -> {Tooltip -> Column[{"Conclusion 1", True}]}, "Hypothesis 1" -> {Tooltip -> Column[{"Hypothesis 1", LeftTee[ LeftTee[ LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]]], \ $CellContext`s], $CellContext`s], $CellContext`k] == LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[ LeftTee[$CellContext`s, $CellContext`s], LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]]], \ $CellContext`k]]], LeftTee[ LeftTee[$CellContext`k, LeftTee[$CellContext`k, LeftTee[ LeftTee[ LeftTee[$CellContext`s, $CellContext`s], LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]]], \ $CellContext`k]]], LeftTee[ LeftTee[ LeftTee[$CellContext`s, $CellContext`s], LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]], LeftTee[$CellContext`k, LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]]]]]}]}, "Substitution Lemma 9" -> { Tooltip -> Column[{"Substitution Lemma 9", LeftTee[ LeftTee[ LeftTee[$CellContext`s, $CellContext`s], LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]], LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]] == LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]], LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]]}]}, "Substitution Lemma 11" -> { Tooltip -> Column[{"Substitution Lemma 11", LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]], LeftTee[ LeftTee[$CellContext`k, LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]], LeftTee[ LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k], LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]]]] == LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]], LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]]}]}, "Substitution Lemma 6" -> { Tooltip -> Column[{"Substitution Lemma 6", LeftTee[ LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]]], LeftTee[ LeftTee[$CellContext`s, $CellContext`s], LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]]]], \ $CellContext`k] == LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]], LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]]}]}, "Substitution Lemma 1" -> { Tooltip -> Column[{"Substitution Lemma 1", LeftTee[ LeftTee[ LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]]], \ $CellContext`s], $CellContext`s], $CellContext`k] == LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[ LeftTee[$CellContext`s, $CellContext`s], LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]]], \ $CellContext`k]]], LeftTee[$CellContext`k, LeftTee[ LeftTee[ LeftTee[$CellContext`s, $CellContext`s], LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]]], \ $CellContext`k]]]}]}}, EdgeStyle -> { DirectedEdge["Substitution Lemma 6", "Substitution Lemma 7"] -> RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]], DirectedEdge["Axiom 1", "Substitution Lemma 1"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Axiom 2", "Substitution Lemma 9"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Axiom 2", "Substitution Lemma 4"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Hypothesis 1", "Substitution Lemma 1"] -> RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]], DirectedEdge["Substitution Lemma 2", "Substitution Lemma 3"] -> RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]], DirectedEdge["Substitution Lemma 1", "Substitution Lemma 2"] -> RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]], DirectedEdge["Substitution Lemma 7", "Substitution Lemma 8"] -> RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]], DirectedEdge["Substitution Lemma 3", "Substitution Lemma 4"] -> RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]], DirectedEdge["Axiom 2", "Substitution Lemma 2"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Substitution Lemma 8", "Substitution Lemma 9"] -> RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]], DirectedEdge["Axiom 2", "Substitution Lemma 6"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Axiom 2", "Substitution Lemma 3"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Axiom 2", "Substitution Lemma 10"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Substitution Lemma 9", "Substitution Lemma 10"] -> RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]], DirectedEdge["Substitution Lemma 10", "Substitution Lemma 11"] -> RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]], DirectedEdge["Axiom 1", "Conclusion 1"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Axiom 2", "Substitution Lemma 5"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Axiom 2", "Substitution Lemma 8"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Substitution Lemma 5", "Substitution Lemma 6"] -> RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]], DirectedEdge["Axiom 2", "Substitution Lemma 11"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Substitution Lemma 4", "Substitution Lemma 5"] -> RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]], DirectedEdge["Axiom 2", "Substitution Lemma 7"] -> Directive[ Dashing[{Small, Small}], RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]], DirectedEdge["Substitution Lemma 11", "Conclusion 1"] -> RGBColor[ Rational[167, 255], Rational[167, 255], Rational[167, 255]]}, GraphLayout -> "LayeredDigraphEmbedding", VertexLabels -> {None}, VertexShapeFunction -> { "Axiom 1" -> "FiveDown", "Substitution Lemma 4" -> "Circle", "Substitution Lemma 6" -> "Circle", "Axiom 2" -> "FiveDown", "Substitution Lemma 8" -> "Circle", "Substitution Lemma 5" -> "Circle", "Conclusion 1" -> "Square", "Substitution Lemma 1" -> "Circle", "Hypothesis 1" -> "Diamond", "Substitution Lemma 10" -> "Circle", "Substitution Lemma 9" -> "Circle", "Substitution Lemma 7" -> "Circle", "Substitution Lemma 11" -> "Circle", "Substitution Lemma 3" -> "Circle", "Substitution Lemma 2" -> "Circle"}, VertexSize -> {{"Scaled", 0.027819888974716114`}}, VertexStyle -> {"Substitution Lemma 8" -> Directive[ RGBColor[ Rational[15, 17], Rational[52, 85], Rational[12, 85]], EdgeForm[]], "Substitution Lemma 7" -> Directive[ RGBColor[ Rational[15, 17], Rational[52, 85], Rational[12, 85]], EdgeForm[]], "Hypothesis 1" -> Directive[ RGBColor[ Rational[146, 255], Rational[10, 17], 0], EdgeForm[]], "Substitution Lemma 6" -> Directive[ RGBColor[ Rational[15, 17], Rational[52, 85], Rational[12, 85]], EdgeForm[]], "Substitution Lemma 10" -> Directive[ RGBColor[ Rational[15, 17], Rational[52, 85], Rational[12, 85]], EdgeForm[]], "Axiom 1" -> Directive[ RGBColor[ Rational[71, 255], Rational[182, 255], Rational[109, 255]], EdgeForm[]], "Substitution Lemma 3" -> Directive[ RGBColor[ Rational[15, 17], Rational[52, 85], Rational[12, 85]], EdgeForm[]], "Substitution Lemma 1" -> Directive[ RGBColor[ Rational[15, 17], Rational[52, 85], Rational[12, 85]], EdgeForm[]], "Substitution Lemma 11" -> Directive[ RGBColor[ Rational[15, 17], Rational[52, 85], Rational[12, 85]], EdgeForm[]], "Conclusion 1" -> Directive[ RGBColor[ Rational[13, 15], Rational[1, 15], 0], EdgeForm[]], "Substitution Lemma 2" -> Directive[ RGBColor[ Rational[15, 17], Rational[52, 85], Rational[12, 85]], EdgeForm[]], "Substitution Lemma 5" -> Directive[ RGBColor[ Rational[15, 17], Rational[52, 85], Rational[12, 85]], EdgeForm[]], "Axiom 2" -> Directive[ RGBColor[ Rational[71, 255], Rational[182, 255], Rational[109, 255]], EdgeForm[]], "Substitution Lemma 9" -> Directive[ RGBColor[ Rational[15, 17], 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TagBox[ TooltipBox[ {RGBColor[ NCache[ Rational[15, 17], 0.8823529411764706], NCache[ Rational[52, 85], 0.611764705882353], NCache[ Rational[12, 85], 0.1411764705882353]], EdgeForm[None], DiskBox[{-3.8483205659035775, 11.}, 0.21728027885656223]}, TagBox[ GridBox[{{"\"Substitution Lemma 1\""}, { RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], ")"}]}], "\[LeftTee]", "s"}], "\[LeftTee]", "s"}], "\[LeftTee]", "k"}], ")"}], "\[Equal]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"k", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"s", "\[LeftTee]", "s"}], "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], ")"}]}], "\[LeftTee]", "k"}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"k", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"s", "\[LeftTee]", "s"}], "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], ")"}]}], "\[LeftTee]", "k"}], ")"}]}], ")"}]}], ")"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 1", LeftTee[ LeftTee[ LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]]], $CellContext`s], \ $CellContext`s], $CellContext`k] == LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[ LeftTee[$CellContext`s, $CellContext`s], LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]]], \ $CellContext`k]]], LeftTee[$CellContext`k, LeftTee[ LeftTee[ LeftTee[$CellContext`s, $CellContext`s], LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]]], \ $CellContext`k]]]}], "Tooltip"]& ], TagBox[ TooltipBox[ {RGBColor[ NCache[ Rational[15, 17], 0.8823529411764706], NCache[ Rational[52, 85], 0.611764705882353], NCache[ Rational[12, 85], 0.1411764705882353]], EdgeForm[None], DiskBox[{-5.479970396812661, 10.}, 0.21728027885656223]}, TagBox[ GridBox[{{"\"Substitution Lemma 2\""}, { RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "s"}], "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "k"}], ")"}], "\[Equal]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"k", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"s", "\[LeftTee]", "s"}], "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], ")"}]}], "\[LeftTee]", "k"}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"k", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"s", "\[LeftTee]", "s"}], "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], ")"}]}], "\[LeftTee]", "k"}], ")"}]}], ")"}]}], ")"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 2", LeftTee[ LeftTee[ LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`s], LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k] == LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[ LeftTee[$CellContext`s, $CellContext`s], LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]]], \ $CellContext`k]]], LeftTee[$CellContext`k, LeftTee[ LeftTee[ LeftTee[$CellContext`s, $CellContext`s], LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]]], \ $CellContext`k]]]}], "Tooltip"]& ], TagBox[ TooltipBox[ {RGBColor[ NCache[ Rational[15, 17], 0.8823529411764706], NCache[ Rational[52, 85], 0.611764705882353], NCache[ Rational[12, 85], 0.1411764705882353]], EdgeForm[None], DiskBox[{-4.479970396732341, 9.}, 0.21728027885656223]}, TagBox[ GridBox[{{"\"Substitution Lemma 3\""}, { RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "s"}], "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "k"}], ")"}], "\[Equal]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"k", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"s", "\[LeftTee]", "s"}], "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], ")"}]}], "\[LeftTee]", "k"}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"k", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", "k"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "k"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 3", LeftTee[ LeftTee[ LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`s], LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k] == LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[ LeftTee[$CellContext`s, $CellContext`s], LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]]], \ $CellContext`k]]], LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]]}], "Tooltip"]& ], TagBox[ TooltipBox[ {RGBColor[ NCache[ Rational[15, 17], 0.8823529411764706], NCache[ Rational[52, 85], 0.611764705882353], NCache[ Rational[12, 85], 0.1411764705882353]], EdgeForm[None], DiskBox[{-3.4799703967440223, 8.}, 0.21728027885656223]}, TagBox[ GridBox[{{"\"Substitution Lemma 4\""}, { RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "s"}], "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "k"}], ")"}], "\[Equal]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"k", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", "k"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "k"}], ")"}]}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"k", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", "k"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "k"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 4", LeftTee[ LeftTee[ LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`s], LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k] == LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]], LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]]}], "Tooltip"]& ], TagBox[ TooltipBox[ {RGBColor[ NCache[ Rational[15, 17], 0.8823529411764706], NCache[ Rational[52, 85], 0.611764705882353], NCache[ Rational[12, 85], 0.1411764705882353]], EdgeForm[None], DiskBox[{-2.4799703967341316, 7.}, 0.21728027885656223]}, TagBox[ GridBox[{{"\"Substitution Lemma 5\""}, { RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{ RowBox[{"s", "\[LeftTee]", "s"}], "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], ")"}]}], "\[LeftTee]", "k"}], ")"}], "\[Equal]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"k", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", "k"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "k"}], ")"}]}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"k", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", "k"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "k"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 5", LeftTee[ LeftTee[ LeftTee[ LeftTee[$CellContext`s, $CellContext`s], LeftTee[$CellContext`s, $CellContext`s]], LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]]], $CellContext`k] == LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]], LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]]}], "Tooltip"]& ], TagBox[ TooltipBox[ {RGBColor[ NCache[ Rational[15, 17], 0.8823529411764706], NCache[ Rational[52, 85], 0.611764705882353], NCache[ Rational[12, 85], 0.1411764705882353]], EdgeForm[None], DiskBox[{-1.4799703967542541, 6.}, 0.21728027885656223]}, TagBox[ GridBox[{{"\"Substitution Lemma 6\""}, { RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", "s"}], "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], ")"}]}], ")"}]}], "\[LeftTee]", "k"}], ")"}], "\[Equal]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"k", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", "k"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "k"}], ")"}]}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"k", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", "k"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "k"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 6", LeftTee[ LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]]], LeftTee[ LeftTee[$CellContext`s, $CellContext`s], LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]]]], $CellContext`k] == LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]], LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]]}], "Tooltip"]& ], TagBox[ TooltipBox[ {RGBColor[ NCache[ Rational[15, 17], 0.8823529411764706], NCache[ Rational[52, 85], 0.611764705882353], NCache[ Rational[12, 85], 0.1411764705882353]], EdgeForm[None], DiskBox[{-0.47997039675757946, 5.}, 0.21728027885656223]}, TagBox[ GridBox[{{"\"Substitution Lemma 7\""}, { RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "k"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"s", "\[LeftTee]", "s"}], "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], ")"}]}], "\[LeftTee]", "k"}], ")"}]}], ")"}], "\[Equal]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"k", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", "k"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "k"}], ")"}]}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"k", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", "k"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "k"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 7", LeftTee[ LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k], LeftTee[ LeftTee[ LeftTee[$CellContext`s, $CellContext`s], LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]]], $CellContext`k]] == LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]], LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], \ $CellContext`k]]]]}], "Tooltip"]& ], TagBox[ TooltipBox[ {RGBColor[ NCache[ Rational[15, 17], 0.8823529411764706], NCache[ Rational[52, 85], 0.611764705882353], NCache[ Rational[12, 85], 0.1411764705882353]], EdgeForm[None], DiskBox[{0.5200296040812873, 4.}, 0.21728027885656223]}, TagBox[ GridBox[{{"\"Substitution Lemma 8\""}, { RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "k"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", "k"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "k"}], ")"}]}], ")"}]}], ")"}], "\[Equal]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"k", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", "k"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "k"}], ")"}]}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"k", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", "k"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"s", "\[LeftTee]", RowBox[{"(", RowBox[{"s", "\[LeftTee]", "s"}], ")"}]}], "\[LeftTee]", "k"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 8", LeftTee[ LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k], LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, $CellContext`s]], $CellContext`k]]] == LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`k, LeftTee[ LeftTee[$CellContext`s, $CellContext`k], LeftTee[ LeftTee[$CellContext`s, LeftTee[$CellContext`s, 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\[FormalI]], LeftTee[\[FormalK], \[FormalI]]]], \[FormalX]], \ \[FormalY]] == LeftTee[ LeftTee[\[FormalX], \[FormalY]], LeftTee[\[FormalX], \[FormalY]]]}], "Tooltip"]& ], TagBox[ TooltipBox[ DiskBox[{-1., 4.}, 0.14851257661299183`], TagBox[ GridBox[{{"\"Substitution Lemma 8\""}, { RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], "\[LeftTee]", "\[FormalI]"}], "\[LeftTee]", "\[FormalX]"}], "\[LeftTee]", "\[FormalY]"}], ")"}], "\[Equal]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalX]", "\[LeftTee]", "\[FormalY]"}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalX]", "\[LeftTee]", "\[FormalY]"}], ")"}]}], ")"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 8", LeftTee[ LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalS]], \[FormalI]], \ \[FormalX]], \[FormalY]] == LeftTee[ 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\[FormalY]] == LeftTee[ LeftTee[\[FormalX], \[FormalY]], LeftTee[\[FormalX], \[FormalY]]]], ForAll[{\[FormalX], \[FormalY]}, LeftTee[ LeftTee[ LeftTee[ LeftTee[\[FormalS], LeftTee[ LeftTee[\[FormalK], \[FormalS]], \[FormalI]]], LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalK]], \[FormalK]], \[FormalI]]], \ \[FormalY]], \[FormalX]] == LeftTee[ LeftTee[\[FormalY], \[FormalX]], LeftTee[\[FormalY], \[FormalX]]]]], And[ ForAll[{\[FormalX], \[FormalY], \[FormalZ]}, LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalX]], \[FormalY]], \[FormalZ]] == LeftTee[ LeftTee[\[FormalX], \[FormalZ]], LeftTee[\[FormalY], \[FormalZ]]]], ForAll[{\[FormalX], \[FormalY]}, LeftTee[ LeftTee[\[FormalK], \[FormalX]], \[FormalY]] == \[FormalX]], ForAll[{\[FormalX]}, LeftTee[\[FormalI], \[FormalX]] == \[FormalX]]], {{ "Axiom", 1} -> Association["Statement" -> $CellContext`x1 == LeftTee[ LeftTee[\[FormalK], $CellContext`x1], $CellContext`x2], "Proof" -> Association[]], {"Axiom", 2} -> Association[ "Statement" -> 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\[FormalK]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]], \[FormalK]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]], \[FormalK]]]]}], "Tooltip"]& ], TagBox[ TooltipBox[ DiskBox[{-4.479970396732341, 7.500000000676025}, 0.19671632147651769`], TagBox[ GridBox[{{"\"Substitution Lemma 3\""}, { RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalS]"}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}], "\[Equal]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 3", LeftTee[ LeftTee[ LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalS]], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]], \[FormalK]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]]}], "Tooltip"]& ], TagBox[ TooltipBox[ DiskBox[{-3.4799703967440223`, 6.666666667267577}, 0.19671632147651769`], TagBox[ GridBox[{{"\"Substitution Lemma 4\""}, { RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalS]"}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}], "\[Equal]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 4", LeftTee[ LeftTee[ LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalS]], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]]}], "Tooltip"]& ], TagBox[ TooltipBox[ DiskBox[{-2.4799703967341316`, 5.83333333385913}, 0.19671632147651769`], TagBox[ GridBox[{{"\"Substitution Lemma 5\""}, { RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}], "\[Equal]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 5", LeftTee[ LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[\[FormalS], \[FormalS]]], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]], \[FormalK]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]]}], "Tooltip"]& ], TagBox[ TooltipBox[ DiskBox[{-1.4799703967542541`, 5.000000000450683}, 0.19671632147651769`], TagBox[ GridBox[{{"\"Substitution Lemma 6\""}, { RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], ")"}]}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}], "\[Equal]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 6", LeftTee[ LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]], LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]]], \[FormalK]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]]}], "Tooltip"]& ], TagBox[ TooltipBox[ DiskBox[{-0.47997039675757946`, 4.166666667042236}, 0.19671632147651769`], TagBox[ GridBox[{{"\"Substitution Lemma 7\""}, { RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}], "\[Equal]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 7", LeftTee[ LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]], LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]], \[FormalK]]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]]}], "Tooltip"]& ], TagBox[ TooltipBox[ DiskBox[{0.5200296040812873, 3.3333333336337887`}, 0.19671632147651769`], TagBox[ GridBox[{{"\"Substitution Lemma 8\""}, { RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}], "\[Equal]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 8", LeftTee[ LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]]}], "Tooltip"]& ], TagBox[ TooltipBox[ DiskBox[{1.5200296040308672`, 2.5000000002253415`}, 0.19671632147651769`], TagBox[ GridBox[{{"\"Substitution Lemma 9\""}, { RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], ")"}], "\[Equal]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 9", LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]]}], "Tooltip"]& ], TagBox[ TooltipBox[ DiskBox[{2.5200296041249715`, 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LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]]]]]}], "Tooltip"]& ], TagBox[ TooltipBox[ DiskBox[{-3.8483205659035775`, 9.166666667492919}, 0.19671632147651769`], TagBox[ GridBox[{{"\"Substitution Lemma 1\""}, { RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], ")"}]}], "\[LeftTee]", "\[FormalS]"}], "\[LeftTee]", "\[FormalS]"}], "\[LeftTee]", "\[FormalK]"}], ")"}], "\[Equal]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 1", LeftTee[ LeftTee[ LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]], \[FormalS]], \ \[FormalS]], \[FormalK]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]], \[FormalK]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]], \[FormalK]]]]}], "Tooltip"]& ], TagBox[ TooltipBox[ DiskBox[{-5.479970396812661, 8.333333334084472}, 0.19671632147651769`], TagBox[ GridBox[{{"\"Substitution Lemma 2\""}, { RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalS]"}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}], "\[Equal]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 2", LeftTee[ LeftTee[ LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalS]], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]], \[FormalK]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]], \[FormalK]]]]}], "Tooltip"]& ], TagBox[ TooltipBox[ DiskBox[{-4.479970396732341, 7.500000000676025}, 0.19671632147651769`], TagBox[ GridBox[{{"\"Substitution Lemma 3\""}, { RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalS]"}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}], "\[Equal]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 3", LeftTee[ LeftTee[ LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalS]], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]], \[FormalK]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]]}], "Tooltip"]& ], TagBox[ TooltipBox[ DiskBox[{-3.4799703967440223`, 6.666666667267577}, 0.19671632147651769`], TagBox[ GridBox[{{"\"Substitution Lemma 4\""}, { RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalS]"}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}], "\[Equal]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 4", LeftTee[ LeftTee[ LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalS]], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]]}], "Tooltip"]& ], TagBox[ TooltipBox[ DiskBox[{-2.4799703967341316`, 5.83333333385913}, 0.19671632147651769`], TagBox[ GridBox[{{"\"Substitution Lemma 5\""}, { RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}], "\[Equal]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 5", LeftTee[ LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[\[FormalS], \[FormalS]]], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]], \[FormalK]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]]}], "Tooltip"]& ], TagBox[ TooltipBox[ DiskBox[{-1.4799703967542541`, 5.000000000450683}, 0.19671632147651769`], TagBox[ GridBox[{{"\"Substitution Lemma 6\""}, { RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], ")"}]}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}], "\[Equal]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 6", LeftTee[ LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]], LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]]], \[FormalK]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]]}], "Tooltip"]& ], TagBox[ TooltipBox[ DiskBox[{-0.47997039675757946`, 4.166666667042236}, 0.19671632147651769`], TagBox[ GridBox[{{"\"Substitution Lemma 7\""}, { RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}], "\[Equal]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 7", LeftTee[ LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]], LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]], \[FormalK]]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]]}], "Tooltip"]& ], TagBox[ TooltipBox[ DiskBox[{0.5200296040812873, 3.3333333336337887`}, 0.19671632147651769`], TagBox[ GridBox[{{"\"Substitution Lemma 8\""}, { RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}], "\[Equal]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 8", LeftTee[ LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]]}], "Tooltip"]& ], TagBox[ TooltipBox[ DiskBox[{1.5200296040308672`, 2.5000000002253415`}, 0.19671632147651769`], TagBox[ GridBox[{{"\"Substitution Lemma 9\""}, { RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], ")"}], "\[Equal]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 9", LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]]}], "Tooltip"]& ], TagBox[ TooltipBox[ DiskBox[{2.5200296041249715`, 1.6666666668168943`}, 0.19671632147651769`], TagBox[ GridBox[{{"\"Substitution Lemma 10\""}, { RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}], ")"}], "\[Equal]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ 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\[FormalS]]], \[FormalK]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]]}], "Tooltip"]& ], TagBox[ TooltipBox[ DiskBox[{3.520029604096095, 0.8333333334084472}, 0.19671632147651769`], TagBox[ GridBox[{{"\"Substitution Lemma 11\""}, { RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", 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"\[Equal]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 11", LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[ LeftTee[\[FormalK], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[ LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]]]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]]}], "Tooltip"]& ], TagBox[ TooltipBox[ DiskBox[{4.04216242124582, 0.}, 0.19671632147651769`], TagBox[ GridBox[{{"\"Conclusion 1\""}, {"True"}}, 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"\[InvisibleSpace]", TagBox[ TagBox[ RowBox[{ RowBox[{ SubscriptBox["\[ForAll]", RowBox[{"{", RowBox[{"\[FormalX]", ",", "\[FormalY]", ",", "\[FormalZ]"}], "}"}]], RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalX]"}], "\[LeftTee]", "\[FormalY]"}], "\[LeftTee]", "\[FormalZ]"}], ")"}], "\[Equal]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalX]", "\[LeftTee]", "\[FormalZ]"}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalY]", "\[LeftTee]", "\[FormalZ]"}], ")"}]}], ")"}]}]}], "&&", RowBox[{ SubscriptBox["\[ForAll]", RowBox[{"{", RowBox[{"\[FormalX]", ",", "\[FormalY]"}], "}"}]], RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"\[FormalK]", "\[LeftTee]", "\[FormalX]"}], "\[LeftTee]", "\[FormalY]"}], ")"}], "\[Equal]", "\[FormalX]"}]}]}], Short[#, 1]& ], "SummaryItem"]}], "\[SpanFromLeft]"}}, GridBoxAlignment -> { "Columns" -> {{Left}}, "Rows" -> {{Automatic}}}, AutoDelete -> False, GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, 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LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[\[FormalK], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]]]]], And[ ForAll[{\[FormalX], \[FormalY], \[FormalZ]}, LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalX]], \[FormalY]], \[FormalZ]] == LeftTee[ LeftTee[\[FormalX], \[FormalZ]], LeftTee[\[FormalY], \[FormalZ]]]], ForAll[{\[FormalX], \[FormalY]}, LeftTee[ LeftTee[\[FormalK], \[FormalX]], \[FormalY]] == \[FormalX]]], {{ "Axiom", 1} -> Association["Statement" -> $CellContext`x1 == LeftTee[ LeftTee[\[FormalK], $CellContext`x1], $CellContext`x2], "Proof" -> Association[]], {"Axiom", 2} -> Association["Statement" -> LeftTee[ LeftTee[$CellContext`x1, $CellContext`x2], LeftTee[$CellContext`x3, $CellContext`x2]] == LeftTee[ LeftTee[ LeftTee[\[FormalS], $CellContext`x1], $CellContext`x3], \ $CellContext`x2], "Proof" -> Association[]], {"Hypothesis", 1} -> Association["Statement" -> LeftTee[ LeftTee[ LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]], \[FormalS]], \[FormalS]], \ \[FormalK]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]], \[FormalK]]]], LeftTee[ LeftTee[\[FormalK], LeftTee[\[FormalK], LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]], \[FormalK]]]], LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[\[FormalK], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]]]]], "Proof" -> Association[]], {"SubstitutionLemma", 1} -> Association["Statement" -> LeftTee[ LeftTee[ LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]], \[FormalS]], \[FormalS]], \ \[FormalK]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]], \[FormalK]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]], \[FormalK]]]], "Proof" -> Association[ "Input" -> {"Hypothesis", 1}, "Position" -> {2, 1, 2}, "Construct" -> {"Axiom", 1}, "Orientation" -> -1, "Rule" -> (LeftTee[ LeftTee[\[FormalK], Pattern[$CellContext`x1, Blank[]]], Pattern[$CellContext`x2, Blank[]]] -> $CellContext`x1), "OutputExpression" -> LeftTee[ LeftTee[ LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]], \[FormalS]], \[FormalS]], \ \[FormalK]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]], \[FormalK]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]], \[FormalK]]]]]], { "SubstitutionLemma", 2} -> Association["Statement" -> LeftTee[ LeftTee[ LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalS]], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]], \[FormalK]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]], \[FormalK]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 1}, "Position" -> {1, 1}, "Construct" -> {"Axiom", 2}, "Orientation" -> -1, "Rule" -> (LeftTee[ LeftTee[ LeftTee[\[FormalS], Pattern[$CellContext`x1, Blank[]]], Pattern[$CellContext`x2, Blank[]]], Pattern[$CellContext`x3, Blank[]]] -> LeftTee[ LeftTee[$CellContext`x1, $CellContext`x3], LeftTee[$CellContext`x2, $CellContext`x3]]), "OutputExpression" -> LeftTee[ LeftTee[ LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalS]], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]], \[FormalK]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]], \[FormalK]]]]]], { "SubstitutionLemma", 3} -> Association["Statement" -> LeftTee[ LeftTee[ LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalS]], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]], \[FormalK]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 2}, "Position" -> {2, 2, 2}, "Construct" -> {"Axiom", 2}, "Orientation" -> -1, "Rule" -> (LeftTee[ LeftTee[ LeftTee[\[FormalS], Pattern[$CellContext`x1, Blank[]]], Pattern[$CellContext`x2, Blank[]]], Pattern[$CellContext`x3, Blank[]]] -> LeftTee[ LeftTee[$CellContext`x1, $CellContext`x3], LeftTee[$CellContext`x2, $CellContext`x3]]), "OutputExpression" -> LeftTee[ LeftTee[ LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalS]], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]], \[FormalK]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]]]], { "SubstitutionLemma", 4} -> Association["Statement" -> LeftTee[ LeftTee[ LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalS]], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 3}, "Position" -> {2, 1, 2, 2}, "Construct" -> {"Axiom", 2}, "Orientation" -> -1, "Rule" -> (LeftTee[ LeftTee[ LeftTee[\[FormalS], Pattern[$CellContext`x1, Blank[]]], Pattern[$CellContext`x2, Blank[]]], Pattern[$CellContext`x3, Blank[]]] -> LeftTee[ LeftTee[$CellContext`x1, $CellContext`x3], LeftTee[$CellContext`x2, $CellContext`x3]]), "OutputExpression" -> LeftTee[ LeftTee[ LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalS]], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]]]], { "SubstitutionLemma", 5} -> Association["Statement" -> LeftTee[ LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[\[FormalS], \[FormalS]]], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]], \[FormalK]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 4}, "Position" -> {1, 1}, "Construct" -> {"Axiom", 2}, "Orientation" -> -1, "Rule" -> (LeftTee[ LeftTee[ LeftTee[\[FormalS], Pattern[$CellContext`x1, Blank[]]], Pattern[$CellContext`x2, Blank[]]], Pattern[$CellContext`x3, Blank[]]] -> LeftTee[ LeftTee[$CellContext`x1, $CellContext`x3], LeftTee[$CellContext`x2, $CellContext`x3]]), "OutputExpression" -> LeftTee[ LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[\[FormalS], \[FormalS]]], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]], \[FormalK]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]]]], { "SubstitutionLemma", 6} -> Association["Statement" -> LeftTee[ LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]], LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]]], \[FormalK]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 5}, "Position" -> {1, 1}, "Construct" -> {"Axiom", 2}, "Orientation" -> -1, "Rule" -> (LeftTee[ LeftTee[ LeftTee[\[FormalS], Pattern[$CellContext`x1, Blank[]]], Pattern[$CellContext`x2, Blank[]]], Pattern[$CellContext`x3, Blank[]]] -> LeftTee[ LeftTee[$CellContext`x1, $CellContext`x3], LeftTee[$CellContext`x2, $CellContext`x3]]), "OutputExpression" -> LeftTee[ LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]], LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]]], \[FormalK]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]]]], { "SubstitutionLemma", 7} -> Association["Statement" -> LeftTee[ LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]], LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]], \[FormalK]]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 6}, "Position" -> 1, "Construct" -> {"Axiom", 2}, "Orientation" -> -1, "Rule" -> (LeftTee[ LeftTee[ LeftTee[\[FormalS], Pattern[$CellContext`x1, Blank[]]], Pattern[$CellContext`x2, Blank[]]], Pattern[$CellContext`x3, Blank[]]] -> LeftTee[ LeftTee[$CellContext`x1, $CellContext`x3], LeftTee[$CellContext`x2, $CellContext`x3]]), "OutputExpression" -> LeftTee[ LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]], LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]], \[FormalK]]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]]]], { "SubstitutionLemma", 8} -> Association["Statement" -> LeftTee[ LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 7}, "Position" -> {1, 2}, "Construct" -> {"Axiom", 2}, "Orientation" -> -1, "Rule" -> (LeftTee[ LeftTee[ LeftTee[\[FormalS], Pattern[$CellContext`x1, Blank[]]], Pattern[$CellContext`x2, Blank[]]], Pattern[$CellContext`x3, Blank[]]] -> LeftTee[ LeftTee[$CellContext`x1, $CellContext`x3], LeftTee[$CellContext`x2, $CellContext`x3]]), "OutputExpression" -> LeftTee[ LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]]]], { "SubstitutionLemma", 9} -> Association["Statement" -> LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 8}, "Position" -> 1, "Construct" -> {"Axiom", 2}, "Orientation" -> -1, "Rule" -> (LeftTee[ LeftTee[ LeftTee[\[FormalS], Pattern[$CellContext`x1, Blank[]]], Pattern[$CellContext`x2, Blank[]]], Pattern[$CellContext`x3, Blank[]]] -> LeftTee[ LeftTee[$CellContext`x1, $CellContext`x3], LeftTee[$CellContext`x2, $CellContext`x3]]), "OutputExpression" -> LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]]]], { "SubstitutionLemma", 10} -> Association["Statement" -> LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 9}, "Position" -> 1, "Construct" -> {"Axiom", 2}, "Orientation" -> -1, "Rule" -> (LeftTee[ LeftTee[ LeftTee[\[FormalS], Pattern[$CellContext`x1, Blank[]]], Pattern[$CellContext`x2, Blank[]]], Pattern[$CellContext`x3, Blank[]]] -> LeftTee[ LeftTee[$CellContext`x1, $CellContext`x3], LeftTee[$CellContext`x2, $CellContext`x3]]), "OutputExpression" -> LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]]]], { "SubstitutionLemma", 11} -> Association["Statement" -> LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[ LeftTee[\[FormalK], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[ LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]]]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 10}, "Position" -> {1, 2}, "Construct" -> {"Axiom", 2}, "Orientation" -> -1, "Rule" -> (LeftTee[ LeftTee[ LeftTee[\[FormalS], Pattern[$CellContext`x1, Blank[]]], Pattern[$CellContext`x2, Blank[]]], Pattern[$CellContext`x3, Blank[]]] -> LeftTee[ LeftTee[$CellContext`x1, $CellContext`x3], LeftTee[$CellContext`x2, $CellContext`x3]]), "OutputExpression" -> LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[ LeftTee[\[FormalK], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[ LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]]]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]]]], { "Conclusion", 1} -> Association[ "Statement" -> True, "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 11}, "Position" -> 1, "Construct" -> {"Axiom", 1}, "Orientation" -> -1, "Rule" -> (LeftTee[ LeftTee[\[FormalK], Pattern[$CellContext`x1, Blank[]]], Pattern[$CellContext`x2, Blank[]]] -> $CellContext`x1), "OutputExpression" -> True]]}], Editable->False, SelectWithContents->True, Selectable->False]}], "}"}]], "Output", TaggingRules->{}, CellChangeTimes->{{3.816016422706163*^9, 3.816016426692401*^9}, 3.8160258870984287`*^9, 3.8161028938552756`*^9}, CellLabel->"Out[2]=", CellID->4732012] }, Open ]] 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RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 4", LeftTee[ LeftTee[ LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalS]], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]]}], "Tooltip"]& ], TagBox[ TooltipBox[ DiskBox[{-2.4799703967341316`, 5.83333333385913}, 0.19671632147651769`], TagBox[ GridBox[{{"\"Substitution Lemma 5\""}, { RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}], "\[Equal]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 5", LeftTee[ LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[\[FormalS], \[FormalS]]], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]], \[FormalK]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]]}], "Tooltip"]& ], TagBox[ TooltipBox[ DiskBox[{-1.4799703967542541`, 5.000000000450683}, 0.19671632147651769`], TagBox[ GridBox[{{"\"Substitution Lemma 6\""}, { RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], ")"}]}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}], "\[Equal]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 6", LeftTee[ LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]], LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]]], \[FormalK]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]]}], "Tooltip"]& ], TagBox[ TooltipBox[ DiskBox[{-0.47997039675757946`, 4.166666667042236}, 0.19671632147651769`], TagBox[ GridBox[{{"\"Substitution Lemma 7\""}, { RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}], "\[Equal]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 7", LeftTee[ LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]], LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]], \[FormalK]]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]]}], "Tooltip"]& ], TagBox[ TooltipBox[ DiskBox[{0.5200296040812873, 3.3333333336337887`}, 0.19671632147651769`], TagBox[ GridBox[{{"\"Substitution Lemma 8\""}, { RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}], "\[Equal]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 8", LeftTee[ LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]]}], "Tooltip"]& ], TagBox[ TooltipBox[ DiskBox[{1.5200296040308672`, 2.5000000002253415`}, 0.19671632147651769`], TagBox[ GridBox[{{"\"Substitution Lemma 9\""}, { RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], ")"}], "\[Equal]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 9", LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]]}], "Tooltip"]& ], TagBox[ TooltipBox[ DiskBox[{2.5200296041249715`, 1.6666666668168943`}, 0.19671632147651769`], TagBox[ GridBox[{{"\"Substitution Lemma 10\""}, { RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}], ")"}], "\[Equal]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 10", LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]]}], "Tooltip"]& ], TagBox[ TooltipBox[ DiskBox[{3.520029604096095, 0.8333333334084472}, 0.19671632147651769`], TagBox[ 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DiskBox[{-4.479970396732341, 7.500000000676025}, 0.19671632147651769`], TagBox[ GridBox[{{"\"Substitution Lemma 3\""}, { RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalS]"}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}], "\[Equal]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 3", LeftTee[ LeftTee[ LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalS]], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]], \[FormalK]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]]}], "Tooltip"]& ], TagBox[ TooltipBox[ DiskBox[{-3.4799703967440223`, 6.666666667267577}, 0.19671632147651769`], TagBox[ GridBox[{{"\"Substitution Lemma 4\""}, { RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalS]"}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}], "\[Equal]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 4", LeftTee[ LeftTee[ LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalS]], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]]}], "Tooltip"]& ], TagBox[ TooltipBox[ DiskBox[{-2.4799703967341316`, 5.83333333385913}, 0.19671632147651769`], TagBox[ GridBox[{{"\"Substitution Lemma 5\""}, { RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}], "\[Equal]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 5", LeftTee[ LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[\[FormalS], \[FormalS]]], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]], \[FormalK]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]]}], "Tooltip"]& ], TagBox[ TooltipBox[ DiskBox[{-1.4799703967542541`, 5.000000000450683}, 0.19671632147651769`], TagBox[ GridBox[{{"\"Substitution Lemma 6\""}, { RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], ")"}]}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}], "\[Equal]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 6", LeftTee[ LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]], LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]]], \[FormalK]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]]}], "Tooltip"]& ], TagBox[ TooltipBox[ DiskBox[{-0.47997039675757946`, 4.166666667042236}, 0.19671632147651769`], TagBox[ GridBox[{{"\"Substitution Lemma 7\""}, { RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}], "\[Equal]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 7", LeftTee[ LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]], LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]], \[FormalK]]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]]}], "Tooltip"]& ], TagBox[ TooltipBox[ DiskBox[{0.5200296040812873, 3.3333333336337887`}, 0.19671632147651769`], TagBox[ GridBox[{{"\"Substitution Lemma 8\""}, { RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}], "\[Equal]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 8", LeftTee[ LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]]}], "Tooltip"]& ], TagBox[ TooltipBox[ DiskBox[{1.5200296040308672`, 2.5000000002253415`}, 0.19671632147651769`], TagBox[ GridBox[{{"\"Substitution Lemma 9\""}, { RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], ")"}], "\[Equal]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 9", LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]]}], "Tooltip"]& ], TagBox[ TooltipBox[ DiskBox[{2.5200296041249715`, 1.6666666668168943`}, 0.19671632147651769`], TagBox[ GridBox[{{"\"Substitution Lemma 10\""}, { RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}], ")"}], "\[Equal]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalK]", "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalK]"}], "\[LeftTee]", RowBox[{"(", RowBox[{ RowBox[{"\[FormalS]", "\[LeftTee]", RowBox[{"(", RowBox[{"\[FormalS]", "\[LeftTee]", "\[FormalS]"}], ")"}]}], "\[LeftTee]", "\[FormalK]"}], ")"}]}], ")"}]}], ")"}]}], ")"}]}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> "Column", GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], Annotation[#, Column[{"Substitution Lemma 10", LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], 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LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[ LeftTee[\[FormalK], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[ LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]]]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]]}], "Tooltip"]& ], TagBox[ TooltipBox[ DiskBox[{4.04216242124582, 0.}, 0.19671632147651769`], TagBox[ GridBox[{{"\"Conclusion 1\""}, {"True"}}, 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LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[\[FormalK], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]]]]], And[ ForAll[{\[FormalX], \[FormalY], \[FormalZ]}, LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalX]], \[FormalY]], \[FormalZ]] == LeftTee[ LeftTee[\[FormalX], \[FormalZ]], LeftTee[\[FormalY], \[FormalZ]]]], ForAll[{\[FormalX], \[FormalY]}, LeftTee[ LeftTee[\[FormalK], \[FormalX]], \[FormalY]] == \[FormalX]]], {{ "Axiom", 1} -> Association["Statement" -> $CellContext`x1 == LeftTee[ LeftTee[\[FormalK], $CellContext`x1], $CellContext`x2], "Proof" -> Association[]], {"Axiom", 2} -> Association["Statement" -> LeftTee[ LeftTee[$CellContext`x1, $CellContext`x2], LeftTee[$CellContext`x3, $CellContext`x2]] == LeftTee[ LeftTee[ LeftTee[\[FormalS], $CellContext`x1], $CellContext`x3], \ $CellContext`x2], "Proof" -> Association[]], {"Hypothesis", 1} -> Association["Statement" -> LeftTee[ LeftTee[ LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]], \[FormalS]], \[FormalS]], \ \[FormalK]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]], \[FormalK]]]], LeftTee[ LeftTee[\[FormalK], LeftTee[\[FormalK], LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]], \[FormalK]]]], LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[\[FormalK], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]]]]], "Proof" -> Association[]], {"SubstitutionLemma", 1} -> Association["Statement" -> LeftTee[ LeftTee[ LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]], \[FormalS]], \[FormalS]], \ \[FormalK]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]], \[FormalK]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]], \[FormalK]]]], "Proof" -> Association[ "Input" -> {"Hypothesis", 1}, "Position" -> {2, 1, 2}, "Construct" -> {"Axiom", 1}, "Orientation" -> -1, "Rule" -> (LeftTee[ LeftTee[\[FormalK], Pattern[$CellContext`x1, Blank[]]], Pattern[$CellContext`x2, Blank[]]] -> $CellContext`x1), "OutputExpression" -> LeftTee[ LeftTee[ LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]], \[FormalS]], \[FormalS]], \ \[FormalK]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]], \[FormalK]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]], \[FormalK]]]]]], { "SubstitutionLemma", 2} -> Association["Statement" -> LeftTee[ LeftTee[ LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalS]], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]], \[FormalK]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]], \[FormalK]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 1}, "Position" -> {1, 1}, "Construct" -> {"Axiom", 2}, "Orientation" -> -1, "Rule" -> (LeftTee[ LeftTee[ LeftTee[\[FormalS], Pattern[$CellContext`x1, Blank[]]], Pattern[$CellContext`x2, Blank[]]], Pattern[$CellContext`x3, Blank[]]] -> LeftTee[ LeftTee[$CellContext`x1, $CellContext`x3], LeftTee[$CellContext`x2, $CellContext`x3]]), "OutputExpression" -> LeftTee[ LeftTee[ LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalS]], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]], \[FormalK]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]], \[FormalK]]]]]], { "SubstitutionLemma", 3} -> Association["Statement" -> LeftTee[ LeftTee[ LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalS]], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]], \[FormalK]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 2}, "Position" -> {2, 2, 2}, "Construct" -> {"Axiom", 2}, "Orientation" -> -1, "Rule" -> (LeftTee[ LeftTee[ LeftTee[\[FormalS], Pattern[$CellContext`x1, Blank[]]], Pattern[$CellContext`x2, Blank[]]], Pattern[$CellContext`x3, Blank[]]] -> LeftTee[ LeftTee[$CellContext`x1, $CellContext`x3], LeftTee[$CellContext`x2, $CellContext`x3]]), "OutputExpression" -> LeftTee[ LeftTee[ LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalS]], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]], \[FormalK]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]]]], { "SubstitutionLemma", 4} -> Association["Statement" -> LeftTee[ LeftTee[ LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalS]], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 3}, "Position" -> {2, 1, 2, 2}, "Construct" -> {"Axiom", 2}, "Orientation" -> -1, "Rule" -> (LeftTee[ LeftTee[ LeftTee[\[FormalS], Pattern[$CellContext`x1, Blank[]]], Pattern[$CellContext`x2, Blank[]]], Pattern[$CellContext`x3, Blank[]]] -> LeftTee[ LeftTee[$CellContext`x1, $CellContext`x3], LeftTee[$CellContext`x2, $CellContext`x3]]), "OutputExpression" -> LeftTee[ LeftTee[ LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalS]], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]]]], { "SubstitutionLemma", 5} -> Association["Statement" -> LeftTee[ LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[\[FormalS], \[FormalS]]], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]], \[FormalK]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 4}, "Position" -> {1, 1}, "Construct" -> {"Axiom", 2}, "Orientation" -> -1, "Rule" -> (LeftTee[ LeftTee[ LeftTee[\[FormalS], Pattern[$CellContext`x1, Blank[]]], Pattern[$CellContext`x2, Blank[]]], Pattern[$CellContext`x3, Blank[]]] -> LeftTee[ LeftTee[$CellContext`x1, $CellContext`x3], LeftTee[$CellContext`x2, $CellContext`x3]]), "OutputExpression" -> LeftTee[ LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[\[FormalS], \[FormalS]]], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]], \[FormalK]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]]]], { "SubstitutionLemma", 6} -> Association["Statement" -> LeftTee[ LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]], LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]]], \[FormalK]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 5}, "Position" -> {1, 1}, "Construct" -> {"Axiom", 2}, "Orientation" -> -1, "Rule" -> (LeftTee[ LeftTee[ LeftTee[\[FormalS], Pattern[$CellContext`x1, Blank[]]], Pattern[$CellContext`x2, Blank[]]], Pattern[$CellContext`x3, Blank[]]] -> LeftTee[ LeftTee[$CellContext`x1, $CellContext`x3], LeftTee[$CellContext`x2, $CellContext`x3]]), "OutputExpression" -> LeftTee[ LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]], LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]]], \[FormalK]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]]]], { "SubstitutionLemma", 7} -> Association["Statement" -> LeftTee[ LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]], LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]], \[FormalK]]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 6}, "Position" -> 1, "Construct" -> {"Axiom", 2}, "Orientation" -> -1, "Rule" -> (LeftTee[ LeftTee[ LeftTee[\[FormalS], Pattern[$CellContext`x1, Blank[]]], Pattern[$CellContext`x2, Blank[]]], Pattern[$CellContext`x3, Blank[]]] -> LeftTee[ LeftTee[$CellContext`x1, $CellContext`x3], LeftTee[$CellContext`x2, $CellContext`x3]]), "OutputExpression" -> LeftTee[ LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]], LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]]], \[FormalK]]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]]]], { "SubstitutionLemma", 8} -> Association["Statement" -> LeftTee[ LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 7}, "Position" -> {1, 2}, "Construct" -> {"Axiom", 2}, "Orientation" -> -1, "Rule" -> (LeftTee[ LeftTee[ LeftTee[\[FormalS], Pattern[$CellContext`x1, Blank[]]], Pattern[$CellContext`x2, Blank[]]], Pattern[$CellContext`x3, Blank[]]] -> LeftTee[ LeftTee[$CellContext`x1, $CellContext`x3], LeftTee[$CellContext`x2, $CellContext`x3]]), "OutputExpression" -> LeftTee[ LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]]]], { "SubstitutionLemma", 9} -> Association["Statement" -> LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 8}, "Position" -> 1, "Construct" -> {"Axiom", 2}, "Orientation" -> -1, "Rule" -> (LeftTee[ LeftTee[ LeftTee[\[FormalS], Pattern[$CellContext`x1, Blank[]]], Pattern[$CellContext`x2, Blank[]]], Pattern[$CellContext`x3, Blank[]]] -> LeftTee[ LeftTee[$CellContext`x1, $CellContext`x3], LeftTee[$CellContext`x2, $CellContext`x3]]), "OutputExpression" -> LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalS]], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]]]], { "SubstitutionLemma", 10} -> Association["Statement" -> LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 9}, "Position" -> 1, "Construct" -> {"Axiom", 2}, "Orientation" -> -1, "Rule" -> (LeftTee[ LeftTee[ LeftTee[\[FormalS], Pattern[$CellContext`x1, Blank[]]], Pattern[$CellContext`x2, Blank[]]], Pattern[$CellContext`x3, Blank[]]] -> LeftTee[ LeftTee[$CellContext`x1, $CellContext`x3], LeftTee[$CellContext`x2, $CellContext`x3]]), "OutputExpression" -> LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[ LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]]]], { "SubstitutionLemma", 11} -> Association["Statement" -> LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[ LeftTee[\[FormalK], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[ LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]]]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 10}, "Position" -> {1, 2}, "Construct" -> {"Axiom", 2}, "Orientation" -> -1, "Rule" -> (LeftTee[ LeftTee[ LeftTee[\[FormalS], Pattern[$CellContext`x1, Blank[]]], Pattern[$CellContext`x2, Blank[]]], Pattern[$CellContext`x3, Blank[]]] -> LeftTee[ LeftTee[$CellContext`x1, $CellContext`x3], LeftTee[$CellContext`x2, $CellContext`x3]]), "OutputExpression" -> LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[ LeftTee[\[FormalK], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[ LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]]]] == LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]], LeftTee[\[FormalK], LeftTee[ LeftTee[\[FormalS], \[FormalK]], LeftTee[ LeftTee[\[FormalS], LeftTee[\[FormalS], \[FormalS]]], \[FormalK]]]]]]], { "Conclusion", 1} -> Association[ "Statement" -> True, "Proof" -> Association[ "Input" -> {"SubstitutionLemma", 11}, "Position" -> 1, "Construct" -> {"Axiom", 1}, "Orientation" -> -1, "Rule" -> (LeftTee[ LeftTee[\[FormalK], Pattern[$CellContext`x1, Blank[]]], Pattern[$CellContext`x2, Blank[]]] -> $CellContext`x1), "OutputExpression" -> True]]}], Editable->False, SelectWithContents->True, Selectable->False]], "Output", TaggingRules->{}, CellChangeTimes->{3.816025935168023*^9, 3.8161029033772326`*^9}, CellLabel->"Out[1]=", CellID->14730866] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"mwGraph", 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